Question #588d8

1 Answer
May 22, 2017

Use the identities:

#sin(x+y)=sin(x)cos(y)+cos(x)sin(y)#
#cos(x-y)=cos(x)cos(y)+sin(x)sin(y)#
#sin^2x+cos^2x=1#

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We will start with the LHS:

#sin(x+y)cos(x-y)#

#(sinxcosy+cosxsiny)(cosxcosy+sinxsiny)#

Now use the FOIL method (from the good ol' days back in Algebra 1)

#sinxcosxcos^2y+sin^2xsinycosy+cos^2xsinycosy+sinxcosxsin^2y#

#(sin^2y+cos^2y)(sinxcosx)+(sin^2x+cos^2x)(sinycosy)#

Now use the third identity listed above.

#sinxcosx+sinycosy#

In fact, #sin(x+y)cos(x-y)# does NOT always equal #sin^2x-sin^2y#. It DOES however equal #1/2(sin2x+sin2y)#, if that was somehow what was meant to be written.